LIU Ji-li

College of Physics and Information Engineering,Shanxi Normal University, Linfen 041000, Shanxi,China

Abstract： In the strong interaction regime, effective Schr?dinger cat states (macroscopic quantum states)appear in the extended Bose-Hubbard Model, which is two-hold degenerate ground state, due to the atom-pair tunnelling. In this paper, it is investigated by a effective potential method and a periodic instanton method that effective Schr?dinger cat states and its transitions from classical to quantum behavior.More importantly, by the criterion for a first-order transition derived from the high-order of perturbation theory, we can be obtain complete phase diagram and find the phase boundary between first and second-order transitions. Then we investigate the influence of atom-pair tunneling on the dominant transition process. In the strong interaction region, these transitions,due to the atom-pair tunneling, indicate the existing of atom-pair tunneling and can be of useful in the experiment investigation.

Key words： effective Schr?dinger cat state; quantum-classical transitions

1 Introduction

The tunneling dynamics of some atoms trapped in a double-well has been studied from the weak interaction to strong interaction by the Bose-Hubbard model (BHM)[1].In the strong interacting regime, in order to describe better the tunneling dynamics, the Bose-Hubbard model was extended and its Hamiltonian was written as,

(1)

whereU2corresponding to the superexchange interactions between atoms on neighboring lattices site[2].Due to appearence of a peculiar atom-pair hopping term, the extended Bose-Hubbard model (EBHM) explains very well the reported experimental observation of correlated tunneling Ref[1].In this paper,atom-pair tunneling-induced effective Schr?dinger cat states and quantum-classical transitions are considered in the extended Bose-Hubbard Model. For theN-atom occupation in the double-well, the pseudoangular momentum operators are defined as

with the total angular momentum

so the Hamiltonian Eq. (1) can be rewritten as

(2)

where the parametersK1=U0-U2andK2=2U2.Two important parametersJandK2correspond to hopping and atom-pair tunnelling process in Eq(2), respectively.Eq(2) is equivalent to a LMG model Hamiltonian studied by different methods for problems of spin tunneling[3～5].Recently, the dynamics and energy spectrum of the extended Bose-Hubbard model were investigated in the strong interaction region[5,6].In this paper,Eq(2) is converted into a single-particle Hamiltonian with an effective potential. Comparing with the well-known Bose-Hubbard model (K2=0), the potential have a structural change at a critical value of the parameters. In the strong interaction regime,the effective potential have two “sphalerons”(called in the usual field-theoretical terminology) which correspond to the two-hold degenerate ground states (effective Schr?dinger cat states).We will focus on the degenerate ground states which is given by a small barrier and a large barrier.

According to classical physics, only barrier penetration leading over the barrier can yield a nonzero barrier transition rate, and the decay rate obeys the Arrhenius law at sufficiently high temperature. However, at temperature close to absolute zero, quantum tunneling is relevant andΓ～exp(-S/η) with S the action at zero temperature, which is a purely quantum phenomenon.So it is either tunneling from thermally excited states (“temperature assisted tunneling”) or thermal fluctuations over the barrier (“thermal activity”) that dominates the transition rate at finite temperature. With lowering temperature, the crossover from thermal activity to temperature assisted tunneling appears, which can be understood as a phase transition from the quantum phase to the classical phase (quantum-classical transition) of a physical system.Quantum-classical transitions, the analogy with the Landau theory of phase transitions, are of either first or second-order[7～9].By a periodic instanton method, it is investigated that the transition from classical-dominate to quantum-dominate on decay of the degenerate ground states in the strong interaction regime. Importantly,according to the criterion for a first-order transition, we can obtain a complete phase diagram and find the phase boundary between first-and second-order transitions in EBHM. The influence of atom-pair tunnelling on the transition process is very novel, which results in occurring of first-order transitions. It is these transitions that indicates the existing of atom-pair tunnelling, and can be of useful in the experiment investigation.

This paper is organized as follows:In Sec 2， we study effective Schr?dinger cat states (macroscopic quantumstates) and two different types of barriers for the extended Bose-Hubbard model. In Sec 3, we review the theory and method on quantum-classical transitions, and obtain the criterion for a first-order transition derived from the high-order of perturbation theory, then we deduce some analytical results on transitions of the degenerate ground states which guide the numerical calculations and investigate the property of these transitions due to the atom-pair tunneling in detail. Finally, in Sec 4, we discuss the conclusion.

2 Effective Schr?dinger cat states and two different types of barriers

In this Section, firstly we have semiclassical approximation on the Bose-Hubbard model described by Eq(2),which is useful of understanding of complex quantum systems; secondly we convert Eq (2) into a single-particle.Hamiltonian and obtain the effective potential for the extended Bose-Hubbard model by the form of differential operators of pseudospin. Both two methods are available for investigating on the ground state of large-size quantum systems, and by above two methods, the ground states and two different types of barriers are investigated.

2.1 On semiclassical approximation

In order to describe the physical situation, the spin operator is represented as a classical spin vector[9],

Sx=ssinθcosφ
Sy=ssinθsinφ
Sz=scosθ.

The Hamiltonian Eq(2) with the normalized parameters is rewritten as the minimum value of Hamiltonian Eq(3) corresponds to two effective Schr?dinger cat states.One can know two ground states atθ0=π/2,φ=φ0=±arccos(2J/K2) are two-fold degenerate, due to atom-pair tunneling from Eq(3). From Fig.1,one can see the degenerate spin ground states in the easyxyplane point two side of the positivexdirection,and move towards thexdirection with increasing of the hopping term constantJ(equivalent of decreasing of the interaction strength).To change the direction of the degenerate ground-state spin from one to another,it has to overcome an energy barrier,moving the spin along either pathSor pathL.

H=-2Jsinθcosφ+K1/2cos2θ+K2/2sin2θcos2φ

(3)

Fig.1 (color online) Classical visualization of thexoyeasy-planexeasy axis spin system Eq (2) corresponding to EBHM
圖1 與拓展玻色哈伯德型對應(yīng)的自旋系統(tǒng)的易平面xoy平面和易軸x軸的示意圖

2.2 In an effective potential

Begining with the effective Schr?dinger equationHΦ(φ)=EΦ(φ),we can convert Eq(2) into a single-particle.Hamiltonian with the effective potential. In terms of the spin generating function ofSzrepresentation such as

with the assumption ?=1,the pseudospin operators have the following form of differential operators[2]

(4)

According to Eq(4), and making use of a proper unitary transformation, the Hamiltonian Eq(2) may be converted into

(5)

where the variable mass of the single particle

U(φ)=-α(cosφ+β)2+α+γ

(6)

whereα=[J-K2(1/2+3/(4s))cosφ][J-K2(1/2-1/(4s))cosφ]/(K1-K2sin2φ)-K2(1/4+1/(4s)-1/(4s2));β=(K1-K2sin2φ)(1/2+1/(4s))/ζ;γ=(K1-K2sin2φ)(1/2+1/(4s))2/ζandζ= [J-K2(1/2-1/(4s))cosφ][1-K2/J(1/2+3/(4s))cosφ]-K2(1/4+1/(4s)-1/(4s2))(K1/J-K2/Jsin2φ).

In the strong interaction region

(7)

To sum up above two subsections, two effective Schr?dinger cat states for EBHM are two-fold degenerate ground state, which are given by two different types of barriers.Two different types of barriers are a small barrier atφ=0 and a large barrierφ=π,corresponding to the pathSand pathLin Fig.1 and Fig.2.Without atom-pair tunnelling term, the extended Bose-Hubbard model reduces to be the well-known Bose-Hubbard model of which the ground state is non-degenerate, so the two-hold degenerate ground states of EBHM is induced by atom-pair tunneling.

Fig.2 (color online) Effective potentialsU(φ) of the extended Bose-Hubbard model forK1=1,s=2 000,K2=0.2 (top panel) andK2=0.6 (bottom panel) with differentJ

圖2 拓展玻色-哈伯德模型不同參數(shù)J的有效勢，其中參數(shù)

K1=1,s=2 000,K2=0.2(上圖)，K2=0.6(下圖)

3 Quantum-classical Transitions of Tunnelling Between Effective Schr?dinger Cat States

In this section, we begin with the theory and method on quantum-classical transitions, investigate transitions from classical to quantum behavior of the ground state in detail.

3.1 Theory and method on quantum-classical transitions

(1)Functional-integral approach and the effective free energy theory on quantum-classical transitions

According to the functional-integral approach[7,10],at sufficiently high temperature, the decay rateΓof a system, due to thermal activation, obeys the Arrhenius law?！玡xp[-U/(kbT)] withUthe Energy of the effective single particle at a temperature, whereas at temperature close to absolute zero, quantum tunneling is relevant and?！玡xp(-S/η) withSthe action at zero temperature. According to the effective free energy theory and a semiclassical limit (S??) in Ref[11,12]. the imaginary-time (τ=it) classical actionSof the system corresponds to periodic instantons' action of which the periodτp=?/T. Classical trajectoriesφ(τ) of periodic instantons satisfy E.-L.equation, so have two class of solutions. The first class of solutions correspond to the effective particle in rest at the bottom of the inversed potential,φ(τ)=φ0, with the assumptionkb=1,?= 1, the actionS0of the system is derived

S0=E0/T

(8)

whereE0=U(φ0), and the exponent Boltzmann formula representing a pure thermal activation. The second class of solutions correspond to the periodic motion of the periodic instanton betweenφ1(E) andφ2(E), the periodτpand actionSTare derived

(9)

whereφ1,2(E) are the roots of the equationU(φ)=Eonφ, andEis the energy of the effective single particle.

Fig. 3 (color online) Visualization of Crossover temperature of the first-order phase transition, non-monotonic periodτpas a function of the instantons energy (top panel), and action of the thermodynamic and the periodic instantons as a function of temperature (bottom panel), respectively.

圖3 一階相變臨界溫度示意圖，上圖為瞬子的不單調(diào)的周期與其能量的關(guān)系圖，下圖為熱力學(xué)作用量與周期瞬子的作用量分別和溫度的關(guān)系圖

(2)Quantum-classical transitions and its order

According to the functional-integral approach and the effective free energy theory, one can analyze the temperature dependence ofSbased upon the dependence of the thermon periodτp=?/T.The crossover from classical-dominate to quantum-dominate tunneling can be understood as a phase transition from the quantum phase to the classical phase (quantum-classical transition). By the dependence on T of the action, quantum-classical transitions of a physical system are investigated, the second derivative d2S/dT2can be interpreted as the specific heat of the system[8,12,13].According to the continuity of the second derivative of the action, quantum-classical transitions, the analogy with the Landau theory of phase transitions, are of either first or second-order.

(3)Criterion for a first-order phase transition

The solutions near the sphaleron have information on the order of quantum-classical transitions. If the instanton's periodτp(E) monotonically decrease with increasingE, both the action and its first derivative onTare continuous atT=T0, there exist a smooth second-order transition from the thermal regime atT>T0to thermally assisted tunneling atT

ExpandingM(φ)[Eq(5)] andU(φ) [Eq(6)] at the top of the barrier, one can obtain the high-order of fluctuation equation, and the frequency of oscillations, such the criterion for first-order phase transitions is obtained[14],

8(K1-2K2)J2±ζ1J-ζ2>0

(10)

3.2 Phase diagram on decay rate

In the strong interaction regime[Eq(7)] , the extended Bose-Hubbard model has two effective Schr?dinger cat states given by a small barrier and a large barrier.It is considered that the quantum-classical transition on decay

Fig.4 (color online) Phase diagram of the extended Bose-Hubbard model for small barriers and large barriers withs=2 000,K1=1 圖4 當(dāng)參數(shù)K1=1，s=2 000時，拓展玻色-哈伯德模型小勢壘和大勢壘隧穿的相圖

According to Eq(7,10),the phase diagram Fig.4 on decay rate withs=2 000 is obtained.We can distinguish three different situations in the phase-transition behavior of both barriers.In one region with a smallJand a smallK2, quantum-classical transitions for both types of barriers are second-order phase transitions, in another region which the hopping term coefficientJis small and the coefficientK2>K1/2, quantum-classical transitions for both types of barriers are first-order phase transitions. In the remainder region, quantum-classical transitions for the small barrier are second or first-order phase transitions while all transitions for the large barrier are first-order transitions.

3.3 Second-order transitions on decay for both barriers

In the region of Fig.4 with a smallJand a smallK2, quantum-classical transitions on decay rate for both types of barriers are second-order phase transitions.To investigate the second-order phase transitions, we calculate the periodτp(E) and the activityS(T) of the periodic instantons numerically, and analyze their dependence onK1,K2andJ, then discuss the crossover temperature of phase transitions.

In the limitE=0, the periodic instanton reduces to an vacuum instanton describing ground-state tunneling atzero temperature through the small or the large barrier. From Eq(9),obviously, the large barrier vacuum instanton action is always greater than that of the small barrier for all values of parameters. Corresponding to second-order phase transitions, both the two types of instantons have periods with monotonical change.By numerical calculation,the thermal activity for the large barrier is always greater than that of the small barrier, too. So tunneling through the small barrier always dominates tunneling much more than through the large barrier in the low temperature regime.

(11)

whereξ1=2(K1-2K2)(1+1/2s),ξ2=K1(1+1/s-1/s2).From Eq.(11),we can find:(i)the crossover temperature of phase transitions for both barriers rises with increasing of atom-pair tunneling, (ii) the crossover temperature of phase transitions for small barrier is always higher than one for large barrier with EBHM.

3.4 First-order transitions on decay for both barriers

In the region of Fig.4 which the hopping term coefficientJis very small and the coefficientK2>K1/2 quantum-classical transitions on decay rate for both two types of barriers are first-order phase transitions. We analyze the strength of first-order phase transitions and numerically calculate the crossover temperature of first-order phase transitions.

The following equation quantifies the strength of first-order transitions：

(12)

whereE0=U(φ0) corresponding to the energy of the effective particle in rest at the bottom of the inversed potential andEccorresponding to the energy of the

Fig.5 (color online)Crossover temperatureTcof first-order transitions as a function onK2andJfor small barriers withs=2 000

圖5 當(dāng)參數(shù)s=2 000時，小勢壘隧穿的一階相變臨界溫度Tc和參婁K2,J的關(guān)系

As we known, the crossover temperature of first-order phase transitions is determined by the hopping term coefficientJand the atom-pair tunneling coefficientK2. The numerical result on the crossover temperatureTc[Fig.3]is shown in Fig.5 and Fig.6.For the small barrier, when the coefficientK2have a increasing change fromK2=0.5K1, the crossover temperature have a decreasing change, however, when the hopping term coefficientJhave a increasing change, the crossover temperature always rise. For the large barrier, we can see that the crossover temperature is dominated by the atom-pair tunneling coefficientK2. When the hopping term coefficientJhas a large change, the crossover temperatures hardly change, however, when the coefficientK2has change from 0.5 to 1, the crossover temperatures always decrease, if the coefficientK2has change from 0.193 8 to 1, the crossover temperatures always first increase then decrease, and have a peak, as shown in Fig.6.

Obviously because the large barrier is higher and wider than the small barrier with the same parameter, the thermal activity for the large barrier is always greater than that of the small barrier from zero to the lower one of two crossover temperatures, which is illustrated in the numerical results.So tunneling through the small barrier always dominates tunneling through the large barrier in the low temperature regime, just like in the region which we considered in above subsection.

Fig.6 (color online) Crossover temperature Tcof first-order transitions as a function on K2and J for large barriers with s=2 000圖6 當(dāng)參數(shù) s=2 000 時,大勢壘隧穿的一階相變臨界溫度 Tc 和參數(shù) K2,J的關(guān)系Fig.7 (color online) Period τp (E) of the instanton as a function on energy E for small barriers with K2=0.6, s=2 000圖7 小勢壘隧穿的瞬子周期 τp(E) 與能量E 的關(guān)系,其中參數(shù) k2=0.6, s=2 000

3.5 Order of the transition controlled for both barriers

For the normalized Hamiltonian Eq(2), the hopping term coefficient J and the atom-pair tunnelling coefficientK2can be tuned adiabatically. From Fig.4, we have found a novel feature of the extended Bose-Hubbard model, which quantum-classical transitions can be either first or second order, depending on the hopping term coefficientJand the atom-pair tunnelling coefficientK2.

This prediction can be tested experimentally in super-cold atom like rubidium atoms (87Rb) trapped by magneto-optical lattices[13].Since the strength of the atom-atom interaction can be manipulated using modern experimental techniques such as Feshbach resonance, the atom-pair tunnelling process is not faintly because of existing of long-range atom-atom interaction e.g., the Coulomb potential or dipole-dipole interaction etc. The new prospective regime can be realized by modern experimental techniques in the future.

4 Conclusion

In this paper, the extended Bose-Hubbard model was converted into a giant pseudospin, which is mapped to asingle-particle problem in an effective potential.The potential for EBHM have a structural change at a critical value of its parameters.In the strong interaction regime, the extended Bose-Hubbard model have a periodic potential with two “sphalerons”.Two-hold degenerate ground states are given by two different types of barriers, which are a small barrier atφ=0 and a large barrier atφ=π. At finite temperature, it is either tunneling from thermally excited states (“temperature assisted tunneling”) or thermal fluctuations over the barrier (“thermal activity”) that dominate the decay rate of degenerate ground states.And the crossover from temperature assisted tunneling to thermal activity can be understood as a quantum-classical phase transition.We investigated the transition for both barriers by a functional-integral approach and a effective free energy theory.Importantly, according to the criterion Eq(14) for first-order transitions, we have obtained a complete phase diagram and found the phase boundary between first-and second-order transitions in EBHM.Considering the influence of atom-pair tunnelling in the extended Bose-Hubbard model, we discussed the crossover temperature of phase transitions, strength of first-order phase transitions, and compared tunneling through the small barrier with tunneling through the large barrier.It is these transitions that indicates the existing of atom-pair tunnelling, and can be of useful in the experiment investigation.

Fig.8 (color online) Period of the instantonτp(E) as function on energyEfor large barriers withK2=0.3,s=2 000

圖8 大勢壘隧穿的瞬子周期τp(E)與能量E的關(guān)系，其中參數(shù)K2=0.3,S=2 000